When I graduated from school and entered the "real" world many years ago, one of my first jobs was in the aerospace field. The company that hired me was under contract to design the communication system for the USA's first orbiting satellite. (We were in a frantic race with Russia to be the first in space.) My assignment was to figure out where to locate receiving stations around the world so that our satellite would be in constant communication with Earth throughout its orbit.
The work involved a branch of mathematics that I had not had in school (spherical trigonometry), and I spent many hours deriving and checking the equations and formulas needed to obtain a solution. When completed, I considered it a scientific work of art. However, in my obsession with making sure that all the basic concepts were correct, I spent less time checking the simple arithmetic involved. As a result, the final report contained two small arithmetic errors. (There were no calculators in those days, so arithmetic calculations could be quite tedious.)* While these errors did not materially affect the final conclusions, they were enough to brand the entire report as "riddled with mistakes". What a heart-breaking way to learn the importance of checking one's arithmetic! This happened over 50 years ago, but the lesson learned is still strong in my mind today.
As I moved into other fields of science, it became more and more apparent that checking one's solution was as important as arriving at the solution in the first place. In fact, some of the checks were more elegant than the solutions. But it made sense to make these checks. After all, in the "real" world there is no teacher or answer book to tell you if the answer arrived at is correct. And sometimes even a simple error can be disastrous (causing a guided missile to hit a hospital instead of an enemy bunker, for example).
So when I encounter students who perform multiple mathematical steps and calculations in their head, I am not awed by their brilliance, instead I am struck with the need to teach them the importance of carefully documenting and checking their steps. Showing the important step by step operations in a solution in a clear logical way makes it easier to retrace and check the solution. And if a hastily written nine looks like a seven, or a three like an eight, what good was the time saved if it results in the wrong answer, especially when everything has to be right to get the right answer, but only one thing wrong will give the wrong answer. Methodically recording the important steps in arriving at a solution, and carefully checking results, is not a sign of poor apprehension, it is a sign of maturity.
* Of interest to teachers: You probably remember the slide rule, but did you know that using logarithms to do multiplication and division was common practice in my school days.
John Schwarz